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Principia Mathematica, chapter *117: a false proposition?
I was reading Principia Mathematica of Whitehead and Russell and I have found what I think is a false proposition.
The proposition in question is *117.632
click on the link to see the formula and the "demonstration"
Now the problem is that the relation T is not one-one as the authors claim.
Let k be the class {$alpha$$_1$; $alpha$$_2$; $alpha$$_3$}.
In turn those classes are described as follow:
$alpha$$_1$={ $x_{11}$; $x_{12}$ ; $x_{13}$ }
$alpha$$_2$={ $x_{21}$; $x_{22}$ ; $x_{23}$ }
$alpha$$_3$={ $x_{31}$; $x_{32}$ ; $x_{33}$ }
Now let $rho$ be the class { $x_{11}$; $x_{21}$ ; $x_{31}$ }
and let $sigma$ be the class { $x_{12}$; $x_{22}$ ; $x_{32}$ }
The hypothesis of the proposition *117.632 are satisfied by those classes.
Let u be the class:
($rho$-$alpha$$_2$-$alpha$$_1$) $cup$ ($sigma$ $cap$ $alpha$$_2$) $cup$ {$x_{11}$}
(which is in turn equal to {$x_{31}$; $x_{22}$; $x_{11}$} )
And v the class:
($rho$-$alpha$$_3$-$alpha$$_1$) $cup$ ($sigma$ $cap$ $alpha$$_3$) $cup$ {$x_{11}$}
(which is in turn equal to {$x_{21}$; $x_{32}$; $x_{11}$}).
Now we have uT$x_{11}$ and vT$x_{11}$ and u is different from v.
Thus T is not one-one.
Maybe there is a way to fix the proposition, but so far I have not been able to do so (the proposition is important because it's used in the demonstration of some other theorems).
The demonstration of the authors of the formula (6) is for me totally obscure.
EDIT
Short Glossary:
" k $in$ Cls$^2$ excl " means " k is a class of classes mutually exclusive, such that there are no two classes of k which have a member in common."
" k $notin$ 0 $cup$ 1 " means " k is a class with at least 2 elements (in our case those elements are classes)."
" $sigma$$in$Prod'k " means " $sigma$ is a class obtained "by choosing" (or "by selecting") for each class of k one, and only one, element. "
" $sigma$$cap$$rho$=$Lambda$ " means " $sigma$ and $rho$ have no common members. "
"$rho$-$beta$" means " the class whose elements are elements of $rho$ but not of $beta$ "
" $iota$'x " means "the class whose only element is x, i.e the class {x} "
The relation T of the proposition is the relation which holds between u and x when, for some classes $alpha$ and $beta$ such that both are member of k and $alpha$ is not $beta$ and x is a member of $beta$, we have:
u=( $rho$-$alpha$-$beta$) $cup$ ($sigma$ $cap$ $alpha$) $cup$ $iota$'x
Principia Mathematica is avaiable for free on archive.org, so everybody can check what I have said.
logic math-history foundations alternative-set-theories
asked Nov 21 '18 at 18:13
Thomas FerrariThomas Ferrari
113
|
show 7 more comments
I was reading Principia Mathematica of Whitehead and Russell and I have found what I think is a false proposition.
The proposition in question is *117.632
click on the link to see the formula and the "demonstration"
Now the problem is that the relation T is not one-one as the authors claim.
Let k be the class {$alpha$$_1$; $alpha$$_2$; $alpha$$_3$}.
In turn those classes are described as follow:
$alpha$$_1$={ $x_{11}$; $x_{12}$ ; $x_{13}$ }
$alpha$$_2$={ $x_{21}$; $x_{22}$ ; $x_{23}$ }
$alpha$$_3$={ $x_{31}$; $x_{32}$ ; $x_{33}$ }
Now let $rho$ be the class { $x_{11}$; $x_{21}$ ; $x_{31}$ }
and let $sigma$ be the class { $x_{12}$; $x_{22}$ ; $x_{32}$ }
The hypothesis of the proposition *117.632 are satisfied by those classes.
Let u be the class:
($rho$-$alpha$$_2$-$alpha$$_1$) $cup$ ($sigma$ $cap$ $alpha$$_2$) $cup$ {$x_{11}$}
(which is in turn equal to {$x_{31}$; $x_{22}$; $x_{11}$} )
And v the class:
($rho$-$alpha$$_3$-$alpha$$_1$) $cup$ ($sigma$ $cap$ $alpha$$_3$) $cup$ {$x_{11}$}
(which is in turn equal to {$x_{21}$; $x_{32}$; $x_{11}$}).
Now we have uT$x_{11}$ and vT$x_{11}$ and u is different from v.
Thus T is not one-one.
Maybe there is a way to fix the proposition, but so far I have not been able to do so (the proposition is important because it's used in the demonstration of some other theorems).
The demonstration of the authors of the formula (6) is for me totally obscure.
EDIT
Short Glossary:
" k $in$ Cls$^2$ excl " means " k is a class of classes mutually exclusive, such that there are no two classes of k which have a member in common."
" k $notin$ 0 $cup$ 1 " means " k is a class with at least 2 elements (in our case those elements are classes)."
" $sigma$$in$Prod'k " means " $sigma$ is a class obtained "by choosing" (or "by selecting") for each class of k one, and only one, element. "
" $sigma$$cap$$rho$=$Lambda$ " means " $sigma$ and $rho$ have no common members. "
"$rho$-$beta$" means " the class whose elements are elements of $rho$ but not of $beta$ "
" $iota$'x " means "the class whose only element is x, i.e the class {x} "
The relation T of the proposition is the relation which holds between u and x when, for some classes $alpha$ and $beta$ such that both are member of k and $alpha$ is not $beta$ and x is a member of $beta$, we have:
u=( $rho$-$alpha$-$beta$) $cup$ ($sigma$ $cap$ $alpha$) $cup$ $iota$'x
Principia Mathematica is avaiable for free on archive.org, so everybody can check what I have said.
logic math-history foundations alternative-set-theories
asked Nov 21 '18 at 18:13
Thomas FerrariThomas Ferrari
113
1
Well... If that book is broken, I'm gonna quit Mathematics!
– Federico
Nov 21 '18 at 18:38
Maybe there is just a typographical error or a missing hypothesis that went unnoticied.
– Thomas Ferrari
Nov 21 '18 at 18:42
1
@Federico: Principia Mathematica is almost certainly riddled with errors (as I am sure Russell and Whitehead would have agreed). However that's no reason to quit mathematics. If you really care about the formal foundations, these days we have technology that let's us verify this kind of detailed formal reasoning mechanically and what is covered in the Principia and a great deal more of mathematics has been mechanically verified.
– Rob Arthan
Nov 21 '18 at 22:29
2
Thomas: you are highly unlikely to find anyone on MSE (or anywhere else) with a sufficiently detailed grasp of the minutiae of Russell and Whitehead's to answer you question off the top of their head. If you can paraphrase the problem you have encountered in ordinary mathematical language, then we may well be able to provide either an informal explanation or an explanation in terms of modern formal type theory. The link to a scan of a proof in Prinicipia Mathematica is not useful to anybody who does not actually have a copy of the text.
– Rob Arthan
Nov 21 '18 at 22:39
I'm difficulty to follow the proof... But what is $text {Prod}' kappa$ ? It is defined in terms of the "arithmetical product" of the classes of $kappa$. If the class $kappa$ has 3 elements with three elements each, the arith product must have 27 elements. See Vol.II, page 67.
– Mauro ALLEGRANZA
Nov 22 '18 at 8:49
|
show 7 more comments
I was reading Principia Mathematica of Whitehead and Russell and I have found what I think is a false proposition.
The proposition in question is *117.632
click on the link to see the formula and the "demonstration"
Now the problem is that the relation T is not one-one as the authors claim.
Let k be the class {$alpha$$_1$; $alpha$$_2$; $alpha$$_3$}.
In turn those classes are described as follow:
$alpha$$_1$={ $x_{11}$; $x_{12}$ ; $x_{13}$ }
$alpha$$_2$={ $x_{21}$; $x_{22}$ ; $x_{23}$ }
$alpha$$_3$={ $x_{31}$; $x_{32}$ ; $x_{33}$ }
Now let $rho$ be the class { $x_{11}$; $x_{21}$ ; $x_{31}$ }
and let $sigma$ be the class { $x_{12}$; $x_{22}$ ; $x_{32}$ }
The hypothesis of the proposition *117.632 are satisfied by those classes.
Let u be the class:
($rho$-$alpha$$_2$-$alpha$$_1$) $cup$ ($sigma$ $cap$ $alpha$$_2$) $cup$ {$x_{11}$}
(which is in turn equal to {$x_{31}$; $x_{22}$; $x_{11}$} )
And v the class:
($rho$-$alpha$$_3$-$alpha$$_1$) $cup$ ($sigma$ $cap$ $alpha$$_3$) $cup$ {$x_{11}$}
(which is in turn equal to {$x_{21}$; $x_{32}$; $x_{11}$}).
Now we have uT$x_{11}$ and vT$x_{11}$ and u is different from v.
Thus T is not one-one.
Maybe there is a way to fix the proposition, but so far I have not been able to do so (the proposition is important because it's used in the demonstration of some other theorems).
The demonstration of the authors of the formula (6) is for me totally obscure.
EDIT
Short Glossary:
" k $in$ Cls$^2$ excl " means " k is a class of classes mutually exclusive, such that there are no two classes of k which have a member in common."
" k $notin$ 0 $cup$ 1 " means " k is a class with at least 2 elements (in our case those elements are classes)."
" $sigma$$in$Prod'k " means " $sigma$ is a class obtained "by choosing" (or "by selecting") for each class of k one, and only one, element. "
" $sigma$$cap$$rho$=$Lambda$ " means " $sigma$ and $rho$ have no common members. "
"$rho$-$beta$" means " the class whose elements are elements of $rho$ but not of $beta$ "
" $iota$'x " means "the class whose only element is x, i.e the class {x} "
The relation T of the proposition is the relation which holds between u and x when, for some classes $alpha$ and $beta$ such that both are member of k and $alpha$ is not $beta$ and x is a member of $beta$, we have:
u=( $rho$-$alpha$-$beta$) $cup$ ($sigma$ $cap$ $alpha$) $cup$ $iota$'x
Principia Mathematica is avaiable for free on archive.org, so everybody can check what I have said.
logic math-history foundations alternative-set-theories
asked Nov 21 '18 at 18:13
Thomas FerrariThomas Ferrari
113
I was reading Principia Mathematica of Whitehead and Russell and I have found what I think is a false proposition.
The proposition in question is *117.632
click on the link to see the formula and the "demonstration"
Now the problem is that the relation T is not one-one as the authors claim.
Let k be the class {$alpha$$_1$; $alpha$$_2$; $alpha$$_3$}.
In turn those classes are described as follow:
$alpha$$_1$={ $x_{11}$; $x_{12}$ ; $x_{13}$ }
$alpha$$_2$={ $x_{21}$; $x_{22}$ ; $x_{23}$ }
$alpha$$_3$={ $x_{31}$; $x_{32}$ ; $x_{33}$ }
Now let $rho$ be the class { $x_{11}$; $x_{21}$ ; $x_{31}$ }
and let $sigma$ be the class { $x_{12}$; $x_{22}$ ; $x_{32}$ }
The hypothesis of the proposition *117.632 are satisfied by those classes.
Let u be the class:
($rho$-$alpha$$_2$-$alpha$$_1$) $cup$ ($sigma$ $cap$ $alpha$$_2$) $cup$ {$x_{11}$}
(which is in turn equal to {$x_{31}$; $x_{22}$; $x_{11}$} )
And v the class:
($rho$-$alpha$$_3$-$alpha$$_1$) $cup$ ($sigma$ $cap$ $alpha$$_3$) $cup$ {$x_{11}$}
(which is in turn equal to {$x_{21}$; $x_{32}$; $x_{11}$}).
Now we have uT$x_{11}$ and vT$x_{11}$ and u is different from v.
Thus T is not one-one.
Maybe there is a way to fix the proposition, but so far I have not been able to do so (the proposition is important because it's used in the demonstration of some other theorems).
The demonstration of the authors of the formula (6) is for me totally obscure.
EDIT
Short Glossary:
" k $in$ Cls$^2$ excl " means " k is a class of classes mutually exclusive, such that there are no two classes of k which have a member in common."
" k $notin$ 0 $cup$ 1 " means " k is a class with at least 2 elements (in our case those elements are classes)."
" $sigma$$in$Prod'k " means " $sigma$ is a class obtained "by choosing" (or "by selecting") for each class of k one, and only one, element. "
" $sigma$$cap$$rho$=$Lambda$ " means " $sigma$ and $rho$ have no common members. "
"$rho$-$beta$" means " the class whose elements are elements of $rho$ but not of $beta$ "
" $iota$'x " means "the class whose only element is x, i.e the class {x} "
The relation T of the proposition is the relation which holds between u and x when, for some classes $alpha$ and $beta$ such that both are member of k and $alpha$ is not $beta$ and x is a member of $beta$, we have:
u=( $rho$-$alpha$-$beta$) $cup$ ($sigma$ $cap$ $alpha$) $cup$ $iota$'x
Principia Mathematica is avaiable for free on archive.org, so everybody can check what I have said.
logic math-history foundations alternative-set-theories
logic math-history foundations alternative-set-theories
asked Nov 21 '18 at 18:13
Thomas FerrariThomas Ferrari
113
asked Nov 21 '18 at 18:13
Thomas FerrariThomas Ferrari
113
edited Nov 22 '18 at 10:15
Thomas Ferrari
asked Nov 21 '18 at 18:13
Thomas FerrariThomas Ferrari
113
asked Nov 21 '18 at 18:13
Thomas FerrariThomas Ferrari
113
asked Nov 21 '18 at 18:13
Thomas FerrariThomas Ferrari
113
113
1
Well... If that book is broken, I'm gonna quit Mathematics!
– Federico
Nov 21 '18 at 18:38
Maybe there is just a typographical error or a missing hypothesis that went unnoticied.
– Thomas Ferrari
Nov 21 '18 at 18:42
1
@Federico: Principia Mathematica is almost certainly riddled with errors (as I am sure Russell and Whitehead would have agreed). However that's no reason to quit mathematics. If you really care about the formal foundations, these days we have technology that let's us verify this kind of detailed formal reasoning mechanically and what is covered in the Principia and a great deal more of mathematics has been mechanically verified.
– Rob Arthan
Nov 21 '18 at 22:29
2
Thomas: you are highly unlikely to find anyone on MSE (or anywhere else) with a sufficiently detailed grasp of the minutiae of Russell and Whitehead's to answer you question off the top of their head. If you can paraphrase the problem you have encountered in ordinary mathematical language, then we may well be able to provide either an informal explanation or an explanation in terms of modern formal type theory. The link to a scan of a proof in Prinicipia Mathematica is not useful to anybody who does not actually have a copy of the text.
– Rob Arthan
Nov 21 '18 at 22:39
I'm difficulty to follow the proof... But what is $text {Prod}' kappa$ ? It is defined in terms of the "arithmetical product" of the classes of $kappa$. If the class $kappa$ has 3 elements with three elements each, the arith product must have 27 elements. See Vol.II, page 67.
– Mauro ALLEGRANZA
Nov 22 '18 at 8:49
|
show 7 more comments
1
Well... If that book is broken, I'm gonna quit Mathematics!
– Federico
Nov 21 '18 at 18:38
Maybe there is just a typographical error or a missing hypothesis that went unnoticied.
– Thomas Ferrari
Nov 21 '18 at 18:42
1
@Federico: Principia Mathematica is almost certainly riddled with errors (as I am sure Russell and Whitehead would have agreed). However that's no reason to quit mathematics. If you really care about the formal foundations, these days we have technology that let's us verify this kind of detailed formal reasoning mechanically and what is covered in the Principia and a great deal more of mathematics has been mechanically verified.
– Rob Arthan
Nov 21 '18 at 22:29
2
Thomas: you are highly unlikely to find anyone on MSE (or anywhere else) with a sufficiently detailed grasp of the minutiae of Russell and Whitehead's to answer you question off the top of their head. If you can paraphrase the problem you have encountered in ordinary mathematical language, then we may well be able to provide either an informal explanation or an explanation in terms of modern formal type theory. The link to a scan of a proof in Prinicipia Mathematica is not useful to anybody who does not actually have a copy of the text.
– Rob Arthan
Nov 21 '18 at 22:39
I'm difficulty to follow the proof... But what is $text {Prod}' kappa$ ? It is defined in terms of the "arithmetical product" of the classes of $kappa$. If the class $kappa$ has 3 elements with three elements each, the arith product must have 27 elements. See Vol.II, page 67.
– Mauro ALLEGRANZA
Nov 22 '18 at 8:49
1
1
Well... If that book is broken, I'm gonna quit Mathematics!
– Federico
Nov 21 '18 at 18:38
Well... If that book is broken, I'm gonna quit Mathematics!
– Federico
Nov 21 '18 at 18:38
Maybe there is just a typographical error or a missing hypothesis that went unnoticied.
– Thomas Ferrari
Nov 21 '18 at 18:42
Maybe there is just a typographical error or a missing hypothesis that went unnoticied.
– Thomas Ferrari
Nov 21 '18 at 18:42
1
1
@Federico: Principia Mathematica is almost certainly riddled with errors (as I am sure Russell and Whitehead would have agreed). However that's no reason to quit mathematics. If you really care about the formal foundations, these days we have technology that let's us verify this kind of detailed formal reasoning mechanically and what is covered in the Principia and a great deal more of mathematics has been mechanically verified.
– Rob Arthan
Nov 21 '18 at 22:29
@Federico: Principia Mathematica is almost certainly riddled with errors (as I am sure Russell and Whitehead would have agreed). However that's no reason to quit mathematics. If you really care about the formal foundations, these days we have technology that let's us verify this kind of detailed formal reasoning mechanically and what is covered in the Principia and a great deal more of mathematics has been mechanically verified.
– Rob Arthan
Nov 21 '18 at 22:29
2
2
Thomas: you are highly unlikely to find anyone on MSE (or anywhere else) with a sufficiently detailed grasp of the minutiae of Russell and Whitehead's to answer you question off the top of their head. If you can paraphrase the problem you have encountered in ordinary mathematical language, then we may well be able to provide either an informal explanation or an explanation in terms of modern formal type theory. The link to a scan of a proof in Prinicipia Mathematica is not useful to anybody who does not actually have a copy of the text.
– Rob Arthan
Nov 21 '18 at 22:39
Thomas: you are highly unlikely to find anyone on MSE (or anywhere else) with a sufficiently detailed grasp of the minutiae of Russell and Whitehead's to answer you question off the top of their head. If you can paraphrase the problem you have encountered in ordinary mathematical language, then we may well be able to provide either an informal explanation or an explanation in terms of modern formal type theory. The link to a scan of a proof in Prinicipia Mathematica is not useful to anybody who does not actually have a copy of the text.
– Rob Arthan
Nov 21 '18 at 22:39
I'm difficulty to follow the proof... But what is $text {Prod}' kappa$ ? It is defined in terms of the "arithmetical product" of the classes of $kappa$. If the class $kappa$ has 3 elements with three elements each, the arith product must have 27 elements. See Vol.II, page 67.
– Mauro ALLEGRANZA
Nov 22 '18 at 8:49
I'm difficulty to follow the proof... But what is $text {Prod}' kappa$ ? It is defined in terms of the "arithmetical product" of the classes of $kappa$. If the class $kappa$ has 3 elements with three elements each, the arith product must have 27 elements. See Vol.II, page 67.
– Mauro ALLEGRANZA
Nov 22 '18 at 8:49
|
show 7 more comments
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1
Well... If that book is broken, I'm gonna quit Mathematics!
– Federico
Nov 21 '18 at 18:38
Maybe there is just a typographical error or a missing hypothesis that went unnoticied.
– Thomas Ferrari
Nov 21 '18 at 18:42
1
@Federico: Principia Mathematica is almost certainly riddled with errors (as I am sure Russell and Whitehead would have agreed). However that's no reason to quit mathematics. If you really care about the formal foundations, these days we have technology that let's us verify this kind of detailed formal reasoning mechanically and what is covered in the Principia and a great deal more of mathematics has been mechanically verified.
– Rob Arthan
Nov 21 '18 at 22:29
2
Thomas: you are highly unlikely to find anyone on MSE (or anywhere else) with a sufficiently detailed grasp of the minutiae of Russell and Whitehead's to answer you question off the top of their head. If you can paraphrase the problem you have encountered in ordinary mathematical language, then we may well be able to provide either an informal explanation or an explanation in terms of modern formal type theory. The link to a scan of a proof in Prinicipia Mathematica is not useful to anybody who does not actually have a copy of the text.
– Rob Arthan
Nov 21 '18 at 22:39
I'm difficulty to follow the proof... But what is $text {Prod}' kappa$ ? It is defined in terms of the "arithmetical product" of the classes of $kappa$. If the class $kappa$ has 3 elements with three elements each, the arith product must have 27 elements. See Vol.II, page 67.
– Mauro ALLEGRANZA
Nov 22 '18 at 8:49